Mort-Hog
If moral relativism is wrong, I don't wanna be right.
Posts: 4,192
You're right, and I'll explain why. Consider an even less trivial question. What is bigger, the number of real numbers between 0 and 1 or the number of real numbers between 0 and 2?
Clearly, there is an infinite number of real numbers between 0 and 1. All of these numbers can be represented by 1/n where n is some integer 1,2,3,4,... Let's call the set of real numbers between 0 and 1 set X. A set is just a list of elements (numbers, operators, matrices, wookies, anything. in this case, numbers).
Obviously, there is an infinite number of real numbers between 0 and 2 as well. Call this set Y.
To compare two infinite sets, we need to find a rule (called a 'correspondence' in set theory terms) that links set X to set Y. Each element in set X should correspond to an element in set Y by this correspondence.
We can't say that the two sets are 'equally big' purely because they are both infinite sets. Sets, in principle, can 'overlap' and share elements like X and Y do. Sets X and Y are not the same set. Infinite sets can have very different sizes, and there are formally lots of different 'sizes' of infinity.
So we need to find this rule. Every element n in X corresponds to 2n in Y, and every element m in Y corresponds to m/2 in X. For example, 0.4 in X corresponds to 0.8 in Y. We write this rule X -> Y as f(n) = 2n, and the inverse Y -> X as f'(m) = m/2.
This tells us that different numbers in X get mapped onto different numbers in Y. There are no numbers that map onto several other numbers.
This is a one-to-one correspondence and this means the two sets have the same size.
So the number of real numbers between 0 and 1 is equal to the number of real numbers between 0 and 2.
For our problem, the set of number of real numbers between 0 and 1 and the set of positive integers, the issue is the same (and even easier) because the mapping is f(n) = 1/n and there is also one-to-one correspondence. So the two infinite sets are of equal size. This is a non-trivial result because 'infinite sets' can, in principle, have different sizes.
"The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt. " - Bertrand Russell
The Triumph of Stupidity in Mortals and Others 1931-1935
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Ribberium Mempyre!
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